An analogue of the Von Staudt-Clausen theorem

The universal von Staudt theorems

“…The congruences for small values of n follow as in [9] with the strengthened results 2 = 1 -p modp , and a2p a = 1 -p mod p , and a2 = 1 -3p mod p . These follow respectively from the congruences uJ(p(p -1))!)…”

“…Dibag [9] has considered the von Staudt theorems for the Bernoulli numbers which are obtained by setting c_|l, if / = P -1, 1 I 0, otherwise, where p is a given prime.…”

The Universal Von Staudt Theorems

“…By assigning different values to the variables c¡ we obtain generalisations of theorems of Dibag [9], Ray [18], Katz [13], and Hurwitz [12].…”

Bounds of divided universal Bernoulli numbers and universal Kummer congruences

“…We will first get good bounds for the p-adic valuations of the coefficients of the divided universal Bernoulli numberB n n when n is divisible by p − 1. As an application, we give a simple proof to Clarke's 1989 universal von Staudt theorem [4] which generalized the theorems of Dibag [6], Ray [11], Katz [9], and Hurwitz [8]. Finally we establish new universal Kummer congruences modulo p for the remaining case when (p − 1)|n.…”